The Tone. The most sought after aspect.

I've always thought that articulation defined a player as much (or more, sometimes) than the tone quaility alone. Raphael Mendez comes to mind first. I think if we could take recordings and fade into a long tone without hearing the attack, it would be much more difficult to tell anyone apart, especially if the notes were at about the same range and volume. In orchestral recordings, I've sometimes head a note fade in softly and it can be hard to even identify the instrument until the articulation gives it away.

Most trumpet players I've known who were obsessed with their "sound" end up sounding artificial and forced, whereas most people who have a very distinctive sound, are distinctive because they just play as themselves.

Thats a darned good question.
It is amazing that a person can put a vintage Martin Committee trumpet in the hands of Maynard, Miles and Hubbard and it would sound different for each player.
As for sound and your description of what you dispise and like, I grew up just the opposite. When I was a kid, I wanted to be Maynard Ferguson and Doc Severensen and hated Miles Davis. Now that I'm (alot) older I like both sound types and can hear what they are saying on their horns. Basically (although some would disagree) trumpet players can be divided in to two basic groups, dark sounding and bright sounding. A few really gifted players can do both but that's rare. Generally your sound will lean one way or the other. This is what you've discovered with the musicians mentioned.
Now, why does this difference occur? according to Phil Woods (from the DVD NYU Jazz Masterclass series) the particular sound a musician makes has to do with their physiogomy(facial appearance esp. as a reflection of inner character). I would agree with this.
From my perspective, sound is of great importance in that if your sound sucks, whatever you play will follow suit. You can play Moto Perpetuo and if your sound sucks, promptly after you play, you'll hear members of the audience jingling their car keys in anticipation of leaving. In short, there's more to music than notes. That "More" is the special voice you project.
As for me, I'm a bright full spectrum player (which sounds great to my ears). However, I do like playing dark on some songs as I've developed a taste for Hubbard, Miles, Wynton ect..
What I do to darken my sound a little is to use a RingMute which is a simple ring of acoustic foam that attaches to the bell rim of my horn. What it does is to dampen some of the high overtones without making the horn sound muted. I use it on gigs and occasionally people say I sound like Miles (which sucks as I'm sure I sound like myself with the brightness of the sound reduced).
My unmuted sound is closer to Maynard/Doc/Arturo timbre. In other words, hot bright and exciting just like Eddie Van Halen would want it.
Hope this helps.

Each trumpet player is different in his or her own right. As you develop you will create your own style, tone/sound and presentation. The evolution of this comes with maturity and "becoming one" with the trumpet. You will begin to listen to the notes, then you will "hear" and "feel" what you are playing and create your style. There is a major difference between listening and hearing.